Original Formal Context Research Articles

Incremental construction of three-way concept lattice for knowledge discovery in social networks

Three-way concept analysis (3WCA), a combination of three-way decision and formal concept analysis, is widely used in the field of knowledge discovery. Generally, constructing three-way concept lattices requires the original formal context and its complement context simultaneously. Additionally, the existing three-way concept lattice construction algorithms focus on the static formal context, and cannot cope with the dynamic formal context that is an essential representation in social networks. Toward this end, this paper pioneers a novel problem and method for the incremental construction of three-way concept lattice for knowledge discovery in social networks. Aiming to facilitate the construction efficiency, this paper firstly investigates the three-way concept lattice construction for attribute-incremental/object-incremental formal contexts, respectively. Then, the dynamic formal context of a social network can be viewed as a special formal context with both attribute-increment and object-increment. Further, we develop the AE/OE concept lattice incremental construction algorithms, called SNS-AE and SNS-OE. Extensive experiments are conducted on various formal contexts to evaluate the effectiveness of our incremental algorithms. The experimental results demonstrate that our proposed incremental algorithms can significantly decrease the construction time of three-way concept lattice compared to the non-incremental algorithm.

Constructing three-way concept lattice based on the composite of classical lattices

Three-way concept analysis provides a novel model for three-way decisions, and also extends the classical formal concept analysis, which has attracted many attentions in recent years. In order to construct three-way concept lattices, original formal context and its complement context need to be considered simultaneously. In this paper, a new method of constructing attribute-induced three-way (AE) and object-induced three-way (OE) concept lattices is proposed. Firstly, we introduce an AE-oriented composite operator and an OE-oriented composite operator that combine a pair of formal concepts coming from the concept lattices of original formal context and its complement context. Secondly, we define the candidate AE concepts and redundant AE concepts based on the AE-oriented composite operator, and define the candidate OE concepts and redundant OE concepts based on the OE-oriented composite operator, respectively. Finally, we propose an AE lattice construction algorithm and an OE lattice construction algorithm. Theoretical analysis and experimental results demonstrate that the proposed method is simple and effective, and it is a useful supplement to current three-way concept lattice construction methods.

On stability of fuzzy formal concepts over randomized one-sided formal context

We propose a probabilistic approach to the issue of one-sided fuzzy formal concepts stability. The modified Rice–Siff algorithm represents a crisp index how to select the relevant concepts from the set of all one-sided fuzzy formal concepts. We suggest to explore the formal concepts stability affected by the random fluctuation of values in a formal context. We describe the algorithm and study the properties of the concept stability using random variables with the Gaussian normal distribution. In combination with the modified Rice–Siff algorithm, the Gaussian probabilistic index improves the analysis of the most relevant one-sided formal concepts from the original one-sided formal context. The connections to recent works in the related directions are presented.

A data reduction method in formal fuzzy contexts

As a basic operation in data mining and knowledge discovery, data reduction can reduce the volume of the data and simplify the representation of knowledge. In this paper we propose a method of attribute reduction in a formal fuzzy context based on the notion of “one-sided fuzzy concept”. According to the importance of attributes, we classify the attributes into three types: core attributes, relatively necessary attributes and unnecessary attributes, which are also referred to the attribute characteristics. We propose judgment theorems and the corresponding algorithms for computing the three types of attribute sets. Moreover, a straightforward attribute reduction method by virtue of attribute characteristics is formulated. We show that the computation of formal concepts on the reduced data set is made more efficient, and yet produces the same lattice structure and conceptual hierarchy as the ones derived from the original formal context.

Knowledge reduction in formal fuzzy contexts

Knowledge reduction is a basic issue in knowledge representation and data mining. Although various methods have been developed to reduce the size of classical formal contexts, the reduction of formal fuzzy contexts based on fuzzy lattices remains a difficult problem owing to its complicated derivation operators. To address this problem, we propose a general method of knowledge reduction by reducing attributes and objects in formal fuzzy contexts based on the variable threshold concept lattices. Employing the proposed approaches, we remove attributes and objects which are non-essential to the structure of a variable threshold concept lattice, i.e., with a given threshold level, the concept lattice constructed from a reduced formal context is made identical to that constructed from the original formal context. Discernibility matrices and Boolean functions are, respectively, employed to compute the attribute reducts and object reducts of the formal fuzzy contexts, by which all the attribute reducts and object reducts of the formal fuzzy contexts are determined without changing the structure of the lattice.

Attribute reduction based on property pictorial diagram.

This paper mainly studies attribute reduction which keeps the lattice structure in formal contexts based on the property pictorial diagram. Firstly, the property pictorial diagram of a formal context is defined. Based on such diagram, an attribute reduction approach of concept lattice is achieved. Then, through the relation between an original formal context and its complementary context, an attribute reduct of complementary context concept lattice is obtained, which is also based on the property pictorial diagram of the original formal context. Finally, attribute reducts in property oriented concept lattice and object oriented concept lattice can be acquired by the relations of attribute reduction between these two lattices and concept lattice of complementary context. In addition, a detailed illustrative example is presented.

A Novel Incorporate Algorithm of Concept Lattice

With the expansion of the research field, the research object of some original seemingly unrelated properties have been studied together. At this time, the number of attribute in formal context has changed. For the increased attributes, we need to construct a new concept lattice. The existing incremental building algorithms of concept lattice need the original formal context as the basis, with single attribute or a set of attribute of the object to rebuild the concept lattice. They can't effectively utilize these existing concept lattice that have not relation in attributes. Here, the paper presents one new algorithm for incorporating concept lattice based on the existed concept lattices. We can directly build the together lattice from bottom to top by direct product operation on the existed concept lattices and the mapping relation between the direct product lattice of two existed concept lattices and the together lattice. Formal contexts that attribute sets have no intersection are fit for this algorithm.

Applying the JBOS reduction method for relevant knowledge extraction

This work presents results from an experiment used to assess the JBOS (junction based on objects similarity) reduction method. Two reductions were made of a formal context about patients having symptoms in a tuberculosis data base. The first reduction used the knowledge expressed in the original formal context and the second used the knowledge expressed in expert rules. The assessment was made, in the first case, by comparison of the performances of the sets of extracted rules (stem bases) before and after the reduction, and in the second case, by comparison of the performances of the set of extracted rules after reduction with that of the expert rules. The performance in the first case was exactly the same as before reduction. In the second case the performance even improved, showing that the weighting process, besides incorporating the expert knowledge, resulted in rules well adjusted to the knowledge expressed in the original formal context. So, both reductions resulted in rule sets absolutely consistent with the original ones. The expert rules, FCA rules and both set of rules obtained after reduction were used also to classify patients of a validation set. In this case, the results have shown that the performance was the same before and after reduction. Therefore, it was shown that by means of an appropriate attributes weight assignment it is possible, by the JBOS method, to achieve a suitable level of performance in a specific task after reduction.